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The method of lower and upper solution for Hilfer evolution equations with non-instantaneous impulses

Author

Listed:
  • Haide Gou

    (Northwest Normal University)

  • Tianxiang Wang

    (Northwest Normal University)

Abstract

In this article, we study the existence of mild solutions for a class of Hilfer fractional evolution equations with non-instantaneous impulses in ordered Banach spaces. The definition of mild solutions for our problem was given based on a $$C_0$$ C 0 -semigroup $$W(\cdot )$$ W ( · ) generated by the operator $$-A$$ - A and probability density function. By means of monotone iterative technique and the method of lower and upper, the existence of extremal mild solutions between lower and upper mild solutions for nonlinear evolution equation with non-instantaneous impulses is obtained under the situation that the corresponding $$C_0$$ C 0 -semigroup $$W(\cdot )$$ W ( · ) and non-instantaneous impulsive function $$\gamma _k$$ γ k are compact, $$W(\cdot )$$ W ( · ) is not compact and $$\gamma _k$$ γ k is compact, $$W(\cdot )$$ W ( · ) and $$\gamma _k$$ γ k are not compact, respectively. At last, two examples are given to illustrate the abstract results.

Suggested Citation

  • Haide Gou & Tianxiang Wang, 2023. "The method of lower and upper solution for Hilfer evolution equations with non-instantaneous impulses," Indian Journal of Pure and Applied Mathematics, Springer, vol. 54(2), pages 499-523, June.
    • Handle: RePEc:spr:indpam:v:54:y:2023:i:2:d:10.1007_s13226-022-00271-4
    DOI: 10.1007/s13226-022-00271-4
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    References listed on IDEAS

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    1. Gautam, Ganga Ram & Dabas, Jaydev, 2015. "Mild solutions for class of neutral fractional functional differential equations with not instantaneous impulses," Applied Mathematics and Computation, Elsevier, vol. 259(C), pages 480-489.
    2. Abbas, Saïd & Benchohra, Mouffak, 2015. "Uniqueness and Ulam stabilities results for partial fractional differential equations with not instantaneous impulses," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 190-198.
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